A056503 -id:A056503 - cp's OEIS frontend

A025065 Number of palindromic partitions of n.

1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296, 11732, 11732, 14742, 14742, 18460, 18460, 23025, 23025, 28629, 28629

Offset: 0

Clark Kimberling

nonn

That is, the number of partitions of n into parts which can be listed in palindromic order.
Alternatively, number of partitions of n into parts from the set {1,2,4,6,8,10,12,...}. - T. D. Noe, Aug 05 2005
Also, partial sums of A035363.
Also number of partitions of n with at most one part occurring an odd number of times. - Reinhard Zumkeller, Dec 18 2013
The first Mathematica program computes terms of A025065; the second computes the k palindromic partitions of user-chosen n. - Clark Kimberling, Jan 20 2014
a(n) is the number of partitions p of n+1 such that 2*max(p) > n+1. - Clark Kimberling, Apr 20 2014.
From Gus Wiseman, Nov 28 2018: (Start)
Also the number of integer partitions of n + 2 that are the vertex-degrees of some hypertree. For example, the a(6) = 7 partitions of 8 that are the vertex-degrees of some hypertree, together with a realizing hypertree are:
     (41111): {{1,2},{1,3},{1,4},{1,5}}
     (32111): {{1,2},{1,3},{1,4},{2,5}}
     (22211): {{1,2},{1,3},{2,4},{3,5}}
    (311111): {{1,2},{1,3},{1,4,5,6}}
    (221111): {{1,2},{1,3},{2,4,5,6}}
   (2111111): {{1,2},{1,3,4,5,6,7}}
  (11111111): {{1,2,3,4,5,6,7,8}}
(End)
Conjecture: a(n) is the length of maximal initial segment of A308355(n-1) that is identical to row n of A128628, for n >= 2. - Clark Kimberling, May 24 2019
From Gus Wiseman, May 21 2021: (Start)
The Heinz numbers of palindromic partitions are given by A265640. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
Also the number of integer partitions of n with a part greater than or equal to n/2. This is equivalent to Clark Kimberling's final comment above. The Heinz numbers of these partitions are given by A344414. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)     (8)
       (11)  (21)  (22)   (32)   (33)    (43)    (44)
                   (31)   (41)   (42)    (52)    (53)
                   (211)  (311)  (51)    (61)    (62)
                                 (321)   (421)   (71)
                                 (411)   (511)   (422)
                                 (3111)  (4111)  (431)
                                                 (521)
                                                 (611)
                                                 (4211)
                                                 (5111)
                                                 (41111)
Also the number of integer partitions of n with at least n/2 parts. The Heinz numbers of these partitions are given by A344296. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)   (21)   (22)    (221)    (222)     (2221)     (2222)
       (11)  (111)  (31)    (311)    (321)     (3211)     (3221)
                    (211)   (2111)   (411)     (4111)     (3311)
                    (1111)  (11111)  (2211)    (22111)    (4211)
                                     (3111)    (31111)    (5111)
                                     (21111)   (211111)   (22211)
                                     (111111)  (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

			The partitions for the first few values of n are as follows:
n: partitions .......................... number
1: 1 ................................... 1
2: 2 11 ................................ 2
3: 3 111 ............................... 2
4: 4 22 121 1111 ....................... 4
5: 5 131 212 11111 ..................... 4
6: 6 141 33 222 1221 11211 111111 ...... 7
7: 7 151 313 11311 232 21112 1111111 ... 7
From _Reinhard Zumkeller_, Jan 23 2010: (Start)
Partitions into 1,2,4,6,... for the first values of n:
1: 1 ....................................... 1
2: 2 11 .................................... 2
3: 21 111 .................................. 2
4: 4 22 211 1111 ........................... 4
5: 41 221 2111 11111 ....................... 4
6: 6 42 4211 222 2211 21111 111111.......... 7
7: 61 421 42111 2221 22111 211111 1111111 .. 7. (End)

David A. Corneth, Table of n, a(n) for n = 0..10000 (first 101 terms from Reinhard Zumkeller that were corrected by _Georg Fischer_, Jan 20 2019)

Cf. A172033, A004277. - Reinhard Zumkeller, Jan 23 2010
Cf. A004526, A030019, A056503, A147878, A320921, A322136.
The bisections are both A000070.
The ordered version (palindromic compositions) is A016116.
The complement is counted by A233771 and A210249.
The case of palindromic prime signature is A242414.
Palindromic partitions are ranked by A265640, with complement A229153.
The case of palindromic plane trees is A319436.
The multiplicative version (palindromic factorizations) is A344417.
A000569 counts graphical partitions.
A027187 counts partitions of even length, ranked by A028260.
A035363 counts partitions into even parts, ranked by A066207.
A058696 counts partitions of even numbers, ranked by A300061.
A110618 counts partitions with length <= half sum, ranked by A344291.
Cf. A000041, A067538, A143773, A209816, A338914, A338915, A340387, A344296, A344414, A344415, A344416.

a025065 = p (1:[2,4..]) where
   p [] _ = 0
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011

import Data.List (group)
a025065 = length . filter (<= 1) .
                   map (sum . map ((`mod` 2) . length) . group) . ps 1
   where ps x 0 = [[]]
         ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 18 2013

Map[Length[Select[IntegerPartitions[#], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &]] &, Range[40]] (* Peter J. C. Moses, Jan 20 2014 *)
n = 8; Select[IntegerPartitions[n], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &] (* Peter J. C. Moses, Jan 20 2014 *)
CoefficientList[Series[1/((1 - x) Product[1 - x^(2 n), {n, 1, 50}]), {x, 0, 60}], x] (* Clark Kimberling, Mar 14 2014 *)

N=66; q='q+O('q^N); Vec( 1/((1-q)*eta(q^2)) ) \\ Joerg Arndt, Mar 11 2014

a(n) = A000070(A004526(n)). - Reinhard Zumkeller, Jan 23 2010
G.f.: 1/((1-q)*prod(n>=1, 1-q^(2*n))). [Joerg Arndt, Mar 11 2014]
a(2*k+2) = a(2*k) + A000041(k + 1). - David A. Corneth, May 29 2021
a(n) ~ exp(Pi*sqrt(n/3)) / (2*Pi*sqrt(n)). - Vaclav Kotesovec, Nov 16 2021

Edited by N. J. A. Sloane, Dec 29 2007

Prepended a(0)=1, added more terms, Joerg Arndt, Mar 11 2014

A285012 Irregular triangle read by rows: T(n,k) is the number of periodic palindromic structures of length n using exactly k different symbols, 1 <= k <= n/2 + 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 6, 5, 1, 1, 7, 6, 1, 1, 13, 19, 7, 1, 1, 15, 25, 10, 1, 1, 25, 64, 43, 10, 1, 1, 31, 90, 65, 15, 1, 1, 50, 208, 220, 85, 13, 1, 1, 63, 301, 350, 140, 21, 1, 1, 99, 656, 1059, 618, 154, 17, 1, 1, 127, 966, 1701, 1050, 266, 28, 1

Offset: 1

Andrew Howroyd, Apr 07 2017

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.

Equivalently, the number of necklaces, up to permutation of the symbols, which when turned over are unchanged. When comparing with the turned over necklace a rotation is allowed but a permutation of the symbols is not.

			Triangle starts:
1
1   1
1   1
1   3    1
1   3    1
1   6    5     1
1   7    6     1
1  13   19     7     1
1  15   25    10     1
1  25   64    43    10     1
1  31   90    65    15     1
1  50  208   220    85    13    1
1  63  301   350   140    21    1
1  99  656  1059   618   154   17   1
1 127  966  1701  1050   266   28   1
1 197 2035  4803  4065  1488  258  21  1
1 255 3025  7770  6951  2646  462  36  1
1 391 6250 21105 24915 12857 3222 410 26 1
1 511 9330 34105 42525 22827 5880 750 45 1
...
Example for n=6, k=2:
Periodic symmetry means results are either in the form abccba or abcdcb.
There are 3 binary words in the form abccba that start with 0 and contain a 1 which are 001100, 010010, 011110. Of these, 011110 is equivalent to 001100 after rotation.
There are 7 binary words in the form abcdcb that start with 0 and contain a 1 which are 000100, 001010, 001110, 010001, 010101, 011011, 011111. Of these, 011111 is equivalent to 000100, 010001 is equivalent to 001010 and 011011 is equivalent to 010010 from the first set.
There are therefore a total of 7 + 3 - 4 = 6 equivalence classes so T(6,2) = 6.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..2600

Columns 2..6 are A056508, A056509, A056510, A056511, A056512.
Partial row sums include A056503, A056504, A056505, A056506, A056507.
Row sums are A285013.
Cf. A284855, A284826, A284823, A284877, A285037, A304972.

\\ Ach is A304972, Prim is A285037.
Ach(n,k=n) = {my(M=matrix(n, k, n, k, n>=k)); for(n=3, n, for(k=2, k, M[n, k]=k*M[n-2, k] + M[n-2, k-1] + if(k>2, M[n-2, k-2]))); M}
Prim(n,k=n\2+1) = {my(A=Ach(n\2+1,k), S=matrix(n\2+1, k, n, k, stirling(n,k,2))); Mat(vectorv(n, n, sumdiv(n, d, moebius(d)*(S[(n/d+1)\2, ] + S[n/d\2+1, ] + if((n-d)%2, A[(n/d+1)\2, ] + A[n/d\2+1, ]))/if(d%2, 2, 1) )))}
T(n,k=n\2+1) = {my(A=Prim(n,k)); Mat(vectorv(n, n, sumdiv(n, d, A[d, ])))}
{ my(A=T(20)); for(n=1, matsize(A)[1], print(A[n,1..n\2+1])) } \\ Andrew Howroyd, Oct 02 2019

\\ column sequence using above code.
ColSeq(n, k=2) = { Vec(T(n,k)[,k]) } \\ Andrew Howroyd, Oct 02 2019

Column k is inverse Moebius transform of column k of A285037. - Andrew Howroyd, Oct 02 2019

A256217 a(n) = A000011(n) - A256216(n).

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 14, 16, 26, 32, 52, 64, 101, 128, 202, 256, 399, 512, 796, 1024, 1583, 2048, 3162, 4096, 6302, 8192, 12586, 16384, 25124, 32768, 50186, 65536, 100232, 131072, 200266, 262144, 400115, 524288, 799568, 1048576, 1597834, 2097152, 3193438, 4194304, 6382637, 8388608, 12757770, 16777216, 25501370

Offset: 1

N. J. A. Sloane, Mar 26 2015

nonn

Counts contiguously substituted cycloalkane polyols (CSCPs).

Consider a bracelet of n beads, each colored blue on one side and red on the other -- each bead changes color when the bracelet is turned over. Reversal is then the same as swapping the colors. a(n) is the number of colorings that are invariant under reversal, up to rotation and turning over. - Ed Wynn, May 22 2021

			The a(7) = 8 bracelets are rrrrrrr, rrrrrrb, rrrrrbb, rrrrbbb, rrrrbrb, rrrbrrb, rrbbrrb, rrbrbrb. - _Ed Wynn_, May 22 2021
a(12) = A056503(12) + 1. The extra bracelet is rrrbrrbbbrbb. - _Andrew Howroyd_, Jun 25 2021

Seiichi Manyama, Table of n, a(n) for n = 1..6644
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Table 2 (M_c).

Cf. A000011, A053656, A056503, A066314, A256216.

The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

a(n) = A056503(n) for odd n. - Andrew Howroyd, Jun 14 2021

A056513 Number of primitive (period n) periodic palindromic structures using a maximum of two different symbols.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 7, 10, 14, 21, 31, 42, 63, 91, 123, 184, 255, 371, 511, 750, 1015, 1519, 2047, 3030, 4092, 6111, 8176, 12222, 16383, 24486, 32767, 49024, 65503, 98175, 131061, 196308, 262143, 392959, 524223, 785910, 1048575, 1572256, 2097151, 3144702, 4194162

Offset: 0

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.

Number of Lyndon compositions (aperiodic necklaces of positive integers) summing to n that can be rotated to form a palindrome. - Gus Wiseman, Sep 16 2018

			From _Gus Wiseman_, Sep 16 2018: (Start)
The sequence of palindromic Lyndon compositions begins:
  (1)  (2)  (3)  (4)    (5)    (6)      (7)
                 (112)  (113)  (114)    (115)
                        (122)  (1122)   (133)
                               (11112)  (223)
                                        (11113)
                                        (11212)
                                        (11122)
(End)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Row sums of A179317.

Cf. A008965, A025065, A056476, A059966, A242414, A285037, A317085, A317086, A317087, A318731.

(* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1;
c[n_] := c[n] = If[EvenQ[n], 2^(n/2 - 1) + c[n/2], 2^((n - 1)/2)];
a56503[n_] := If[OddQ[n], b[n]/2, (1/2)*(b[n] + c[n/2])];
a[n_] := DivisorSum[n, MoebiusMu[#] a56503[n/#]&];
Array[a, 45] (* Jean-François Alcover, Jun 29 2018, after Andrew Howroyd *)

a(n) = {if(n < 1, n==0, sumdiv(n, d, moebius(d)*(2 + d%2)*(2^(n/d\2)))/(4 - n%2))} \\ Andrew Howroyd, Sep 26 2019

seq(n) = Vec(1 + (1/2)*sum(k=1, n, moebius(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - moebius(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)) + O(x*x^n))) \\ Andrew Howroyd, Sep 27 2019

a(n) = Sum_{d|n} mu(d)*A056503(n/d) for n > 0.
a(n) = Sum_{k=1..2} A285037(n, k). - Andrew Howroyd, Apr 08 2017
G.f.: 1 + (1/2)*Sum_{k>=1} mu(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - mu(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)). - Andrew Howroyd, Sep 27 2019

a(17)-a(45) from Andrew Howroyd, Apr 08 2017

a(0)=1 prepended by Andrew Howroyd, Sep 27 2019

A056509 Number of periodic palindromic structures of length n using exactly three different symbols.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 6, 19, 25, 64, 90, 208, 301, 656, 966, 2035, 3025, 6250, 9330, 19035, 28501, 57740, 86526, 174436, 261625, 525994, 788970, 1583119, 2375101, 4760516, 7141686, 14303011, 21457825, 42954850, 64439010, 128953341, 193448101, 387046700, 580606446, 1161504423

Offset: 1

Marks R. Nester

nonn

			For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column 3 of A285012.

Cf. A000392, A056503, A056504, A056519.

a(n) = A056504(n) - A056503(n).

Inverse Moebius transform of A056519. - Andrew Howroyd, Oct 01 2019

a(17)-a(35) from Andrew Howroyd, Apr 07 2017

Terms a(36) and beyond from Andrew Howroyd, Oct 01 2019

A056508 Number of periodic palindromic structures of length n using exactly two different symbols.

Original entry on oeis.org

0, 1, 1, 3, 3, 6, 7, 13, 15, 25, 31, 50, 63, 99, 127, 197, 255, 391, 511, 777, 1023, 1551, 2047, 3090, 4095, 6175, 8191, 12323, 16383, 24639, 32767, 49221, 65535, 98431, 131071, 196743, 262143, 393471, 524287, 786697, 1048575, 1573375, 2097151, 3146255, 4194303

Offset: 1

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.

For odd n, a palindrome cannot be the complement of itself, so a(n) is given by A284855(n,2)/2 - 1. - Andrew Howroyd, Apr 08 2017

			From _Andrew Howroyd_, Apr 07 2017: (Start)
Example for n=6:
Periodic symmetry means results are either in the form abccba or abcdcb.
There are 3 binary words in the form abccba that start with 0 and contain a 1 which are 001100, 010010, 011110. Of these, 011110 is equivalent to 001100 after rotation.
There are 7 binary words in the form abcdcb that start with 0 and contain a 1 which are 000100, 001010, 001110, 010001, 010101, 011011, 011111. Of these, 011111 is equivalent to 000100, 010001 is equivalent to 001010 and 011011 is equivalent to 010010 from the first set.
There are therefore a total of 7 + 3 - 4 = 6 equivalence classes so a(6) = 6.
(End)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Giovanni Resta, Table of n, a(n) for n = 1..1000

Column 2 of A285012.

Cf. A052551.

(* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1;
c[n_] := c[n] = If[EvenQ[n], 2^(n/2 - 1) + c[n/2], 2^((n - 1)/2)];
a[n_] := If[OddQ[n], b[n]/2, (1/2)*(b[n] + c[n/2])] - 1;
Array[a, 45] (* Jean-François Alcover, Jun 29 2018, after Andrew Howroyd *)

a(n) = A056503(n) - 1.

a(2n + 1) = 2^n - 1. - Andrew Howroyd, Apr 07 2017

a(17)-a(45) from Andrew Howroyd, Apr 07 2017

A179181 'PE(n,k)' triangle read by rows. PE(n,k) is the number of k-palindromes of n up to cyclic equivalence.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 1, 1, 1, 1, 0, 3, 0, 3, 0, 1, 1, 1, 3, 2, 3, 2, 1, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 1, 4, 2, 6, 4, 4, 2, 1, 1, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 1, 1, 5, 3, 10, 6, 10, 5, 5, 3, 1, 1

Offset: 1

John P. McSorley, Jun 30 2010

A k-composition of n is an ordered collection of k positive integers (parts) which sum to n.
Two k-compositions of n are cyclically equivalent if one can be obtained from the other by a cyclic permutation of its parts.
A k-palindrome of n is a k-composition of n which is a palindrome.
Let PE(n,k) denote the number of k-palindromes of n up to cyclic equivalence.
This sequence is the 'PE(n,k)' triangle read by rows.

			The triangle begins
  1
  1,1
  1,0,1
  1,1,1,1
  1,0,2,0,1
  1,1,2,1,1,1
  1,0,3,0,3,0,1
  1,1,3,2,3,2,1,1
  1,0,4,0,6,0,4,0,1
  1,1,4,2,6,4,4,2,1,1
For example, row 8 is 1,1,3,2,3,2,1,1.
We have PE(8,3)=3 because there are 3 3-palindromes of 8, namely: 161, 242, and 323, and none are cyclically equivalent to the others.
We have PE(8,4)=2 because there are 3 4-palindromes of 8, namely: 3113, 1331, and 2222, but 3113 and 1331 are cyclically equivalent.

John P. McSorley: Counting k-compositions with palindromic and related structures. Preprint, 2010.

Andrew Howroyd, Table of n, a(n) for n = 1..1275

If we ignore cyclic equivalence then we have sequence A051159.
The row sums of the 'PE(n, k)' triangle give sequence A056503.
Cf. A179317, A179519.

T[n_, k_] := (3 - (-1)^k)/4*Sum[MoebiusMu[d]*QBinomial[n/d - 1, k/d - 1, -1], {d, Divisors@ GCD[n, k]}]; Table[DivisorSum[GCD[n, k], T[n/#, k/#] &], {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Oct 31 2021, after Jean-François Alcover at A179317 *)

p(n, k) = if(n%2==1&&k%2==0, 0, binomial((n-1)\2, (k-1)\2));
APE(n, k) = if(k%2,1,1/2) * sumdiv(gcd(n,k), d, moebius(d) * p(n/d, k/d));
T(n, k) = sumdiv(gcd(n,k), d, APE(n/d, k/d));
for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) \\ Andrew Howroyd, Oct 07 2017

PE(n, k) = Sum_{d | gcd(n,k)} A179317(n/d, k/d). - Andrew Howroyd, Oct 07 2017

Terms a(56) and beyond from Andrew Howroyd, Oct 07 2017

A322136 Numbers whose number of prime factors counted with multiplicity exceeds half their sum of prime indices by at least 1.

Original entry on oeis.org

4, 8, 12, 16, 24, 32, 36, 40, 48, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 192, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 432, 448, 480, 512, 576, 640, 648, 672, 704, 720, 768, 800, 832, 864, 896, 960, 972

Offset: 1

Gus Wiseman, Nov 27 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence lists all Heinz numbers of integer partitions where the number of parts is at least 1 plus half the sum of parts.

Also Heinz numbers of integer partitions that are the vertex-degrees of some hypertree. We allow no singletons in a hypertree, so 2 is not included.

			The sequence of partitions with Heinz numbers in the sequence begins: (11), (111), (211), (1111), (2111), (11111), (2211), (3111), (21111), (111111), (22111), (31111), (211111), (22211), (41111), (32111), (1111111).

Cf. A000569, A025065, A030019, A056156, A056239, A056503, A112798, A181821, A242414, A304382, A320922, A320923, A320924, A320925.

Select[Range[1000],PrimeOmega[#]>=(Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]+2)/2&]

A344417 Number of palindromic factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, May 22 2021

nonn

A palindrome is a sequence that is the same whether it is read forward or in reverse. A palindromic factorization of n is a finite multiset of positive integers > 1 with product n that can be permuted into a palindrome.

			The palindromic factorizations for n = 2, 4, 16, 36, 64, 144:
  (2)  (4)    (16)       (36)       (64)           (144)
       (2*2)  (4*4)      (6*6)      (8*8)          (12*12)
              (2*2*4)    (2*2*9)    (4*4*4)        (4*4*9)
              (2*2*2*2)  (3*3*4)    (2*2*16)       (4*6*6)
                         (2*2*3*3)  (2*2*4*4)      (2*2*36)
                                    (2*2*2*2*4)    (3*3*16)
                                    (2*2*2*2*2*2)  (2*2*6*6)
                                                   (3*3*4*4)
                                                   (2*2*2*2*9)
                                                   (2*2*3*3*4)
                                                   (2*2*2*2*3*3)

Positions of 1's are A005117.
The case of palindromic compositions is A016116.
The additive version (palindromic partitions) is A025065.
The case of palindromic prime signature is A242414.
The case of palindromic plane trees is A319436.
A001055 counts factorizations.
A229153 ranks non-palindromic partitions.
A265640 ranks palindromic partitions.
Cf. A000041, A000070, A000188, A004526, A030229, A056503, A057567, A082293, A344414.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
palQ[y_]:=Select[Permutations[y],#==Reverse[#]&]!={};
Table[Length[Select[facs[n],palQ]],{n,50}]

a(2^n) = A025065(n).

a(n) = A057567(A000188(n)). - Andrew Howroyd, May 22 2021

cp's OEIS Frontend

A025065 Number of palindromic partitions of n.

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A285012 Irregular triangle read by rows: T(n,k) is the number of periodic palindromic structures of length n using exactly k different symbols, 1 <= k <= n/2 + 1.

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A256217 a(n) = A000011(n) - A256216(n).

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A056513 Number of primitive (period n) periodic palindromic structures using a maximum of two different symbols.

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A056509 Number of periodic palindromic structures of length n using exactly three different symbols.

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A056508 Number of periodic palindromic structures of length n using exactly two different symbols.

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A179181 'PE(n,k)' triangle read by rows. PE(n,k) is the number of k-palindromes of n up to cyclic equivalence.

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